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THE MONGE-AMPE`RE EQUATION WITH GUILLEMIN BOUNDARY CONDITIONS DANIEL RUBIN 4 1 0 2 Abstract. Existence and boundary regularity away from the corners are established n fortwo-dimensionalMonge-AmpèreequationsonconvexpolytopeswithGuilleminbound- a ary conditions. An important step is to derive an expansion in terms of functions yn J and ynlogy for solutions to equations of the form detD2u(x,y)=y−1 in a half-ball. 5 1 1. Introduction ] P A The aim of this paper is to study a seemingly new type of boundary value problem for . a real Monge-Ampère equation in a convex polytope. More precisely, let P ⊂ Rn be a h t polytope, and let a m P = ∩N {l (x) > 0}, (1.1) i=1 i [ be a representation of P as an intersection of half-planes, with l (x) an affine function i 1 of x for each i. We consider the problem of finding a function u ∈ C0(P) and strictly v convex satisfying 7 6 1 7 detD2u(x) = (1.2) 3 ϕ(x) . 1 N 0 u(x)− l (x)logl (x) ∈ C∞(P) (1.3) i i 4 1 Xi=1 : v where the given function ϕ on P is of the form i X N r ϕ(x) = h(x) l (x) (1.4) a i i=1 Y withh(x) ∈ C∞(P),0 < h(x). Boundaryconditionsoftheform(1.3)arecalledGuillemin boundary conditions. The motivation for this problem comes from toric geometry, and particularly Abreu’s equation [A], which is the equation for a Ka¨hler metric of constant scalar curvature on a toric variety. We shall say more about this later, but for the moment, we note that an essential part of the problem is the particular form of the boundary condition, and the fact that the equation takes place on a polytope. For example, a naive version of the problem on a strictly convex domain D ⊂ R2 with boundary function d(x) of the form u − d log d ∈ C3(D), detD2u(x) = O(d−1), would have no solution, since the boundary asymptotics for u would imply that detD2u(x) ∼ d−1log d−1 near ∂D. Thus 1 2 DANIELRUBIN the polytope features of the problem have to be fully taken into account, and they play indeed a major role in our results which we describe next. Let n be the unit, inward-pointing normal to the face l (x) = 0 of the polytope P, and i i set l (x) = n ·x−λ. Any two vectors n and n define a matrix nαnβ, the determinant i i i k i k of which is the area of the parallelogram spanned by n and n . We also denote the N i k vertices of the polytope P by v , 1 ≤ i ≤ N, with v the intersection of the faces l = 0 i i i−1 and l = 0.Then i Theorem 1.1. Let P be a convex polytope in R2, and consider the problem (1.2, 1.3) where h ∈ C∞(P) and h(x) > 0. (a) If the equation admits a solution u which is convex in P and satisfies the boundary condition in (1.2, 1.3), then the given function h(x) must satisfy −1 h(v ) = det(n n )2 l (v ) (1.5) i i−1 i j i ! j6=i−1,i Y (b) Conversely, assume that the given function h(x) satisfies (1.5). Then there exists α > 0 such that for each choice of values {α }N , a ∈ R, there is a unique solution i i=1 i u ∈ Cα(P) to the equation (1.2), satisfying the following boundary condition N u− l (x)logl (x) ∈ C∞(P \{v ,··· ,v }), and u(v ) = α , 1 ≤ α ≤ N. (1.6) i i 1 N i i i=1 X At this moment, the regularity of the solution at the corners is still open. We discuss briefly some of the main steps in the proof of Theorem 1.1. A key obser- vation is that, if a solution u exists, then its restriction to the edge e is the solution of i the following second-order ODE along the edge e , i ∂2 u = |n |2/ϕ (1.7) Ti i ni Combined with the assigned values of u at the vertices v , this equation determines i completelytherestrictionofutotheboundary∂P ofthepolytope. Thus, wecanobtainu by solving the Monge-Ampère equation (1.2) with this given Dirichlet condition. Because the right hand side of the equation (1.2) blows up near the boundary, and because the domain is a polytope, the solution does not appear to have been written down previously in the literature. However, we show in section §3 that the methods of Cheng-Yau [CY] can be suitably extended to produce a generalized solution. The remaining issue is the regularity. The Cα regularity on P is established by con- structing suitable barrier functions. The regularity and asymptotic expansion at the edges are modeled on the following problem detD2u(x′,x ) = x−1 (1.8) n n near the interior of a face {x = 0}, and n detD2u = (x ...x )−1 (1.9) 1 n THE MONGE-AMPE`RE EQUATION WITH GUILLEMIN BOUNDARY CONDITIONS 3 nearacorner. Theequation(1.8)isalimitcaseoftheequationsstudiedbyDaskolopoulos and Savin in [DS] (and in [S] in higher dimensions) of the form detD2v(x,y) = yα in B , α > −1 (1.10) 1 for which they obtained the behavior of the solution 1 a v(x,y) = x2 + |y|2+α+O (x2 +|y|2+α)1+δ (1.11) 2a (α+2)(α+1) (cid:0) (cid:1) for some a > 0, in a neighborhood of the origin. The case of exponent -1 presents a new difficulty from the fact that solutions with quadratic growth on the flat boundary have infinite normal derivative. In our case, we need to combine the techniques of [DS] with a careful analysis of the partial Legendre transform of u and of the Monge-Ampère equation. We say now a few words about the motivation from toric geometry. Let X be a toric variety of dimension n. Then its image under the moment map is a polytope P in Rn, and a toric Ka¨hler metric on X can be determined by a function u : P¯ → Rn called the symplectic potential, which is the Legendre transform of the Ka¨hler potential in the open torus. Guillemin [G] showed that the symplectic potential of a smooth toric variety satisfies N u(x) = l (x)logl (x)+f(x), f ∈ C∞(P), u convex in P (1.12) i i i=1 X where the affine functions l (x) defining the faces of P have been appropriately normal- i ized. As shown by Abreu [A], the Ka¨hler metric is an extremal metric if and only if its symplectic potential u satisfies the so-called Abreu equation n ∂2uij = −A, (1.13) ∂x ∂x i j i,j=1 X where (uij) is the inverse of the Hessian (u ) where A is an affine function. The metric ij is of constant scalar curvature when A is constant. The Abreu’s equation is clearly equivalent to the following system of two second-order elliptic equations for the two unknowns (u,ϕ), detD2u = ϕ−1 (1.14) Uijϕ = −A (1.15) ij where (Uij) is the cofactor matrix of the Hessian of u. From the boundary condition, it follows that the function ϕ(x) must vanish to first order along each face. The existence of a metric of constant scalar curvature, and hence the solvability of Abreu’s equation, has been shown by Donaldson in dimension n = 2 to be equivalent to the K-stability of the toric variety X [D1]. The same statement is expected to hold in all dimensions, and is known as the Tian-Yau-Donaldson conjecture [D2] (see also [PS] for a survey). Donaldson also gave interior estimates for Abreu’s equation, using in part works of Trudinger-Wang [TW] on similar equations arising from the affine Plateau problem. Donaldson’s results were subsequently extended by Chen, Li, and Sheng [CLS], 4 DANIELRUBIN who solved the problem of general prescribed curvatures in dimension two, and also by Chen, Han, Li, and Sheng [CHLS] giving interior estimates for all dimensions. But even in dimension n = 2, a major question is to understand the singularities of the solutionsofAbreu’sequationingeneral. ThisisadifficultproblemsinceAbreu’sequation is of fourth-order, and it is natural as a first step to explore separately the two second- order equations appearing in (1.14). The second equation is a linearized Monge-Ampère equation of the type studied by Caffarelli-Gutierrez [CG]. The first equation, together with the Guillemin boundary conditions, is a new type of boundary value problem for the Monge-Ampère equation, which may be of independent interest and is the equation studied in the present paper. The paper is organized as follows: In section 2, we explain the setup and derive the necessary conditions on the right-hand side, as well as the boundary equation, which we solve to give Dirichlet data compatible with the Guillemin boundary conditions. In section 3, we give a Perrón’s method argument to solve the Dirichlet problem, ensuring thatthereexistsasolutioninthepolytopewhichisHöldercontinuousuptotheboundary. In section 4, the main part of the paper, we deal with the behavior of the solution near an edge, establishing that under the precise boundary relation, the solution goes like l logl + f, with f smooth. This completes the proof of Theorem 1.1. For the most i i part we work exclusively in dimension two. This restriction is mainly for simplicity of computation in sections two and three, where the results have clear extensions to higher dimensions, but is essential in section four where we take the partial Legendre transform of the Monge-Ampère equation to yield a quasilinear equation. Acknowledgements: I would like to thank my advisor D.H. Phong for his guidance andencouragement, andIamalso gratefulto OvidiuSavin andConnor Mooneyformany helpful conversations. 2. Consequences of the Guillemin boundary conditions In general, an asymptotic expansion for the solution u near the boundary of a domain will put some constraints on the boundary behavior of detD2u. In the case of Guillemin boundary conditions on a polytope, these constraints turn out to be quite powerful. This is the contents of Theorem 1.1, part (a), which we reformulate as the following separate proposition for convenience: Lemma 2.1. Let u ∈ C0(P) ∩ C∞(P \ {v ,··· ,v }) be a function which satisfies the 1 N Guillemin boundary condition (1.3) on P \{v ,··· ,v } in the sense that 1 N N u(x)− l (x)logl (x) ∈ C0(P)∩C∞(P \{v ,··· ,v }). (2.1) i i 1 N i=1 X Then 1 detD2u = , (2.2) h(x) N l (x) i=1 i Q THE MONGE-AMPE`RE EQUATION WITH GUILLEMIN BOUNDARY CONDITIONS 5 where h(x) is a function which is in C0(P)∩C∞(P\{v ,··· ,v }) and satisfies 0 < h(x). 1 N When the full Guillemin boundary condition (1.3) holds, then h ∈ C∞(P). Furthermore, 1 h(v ) = . (2.3) k det(n n )2 l (v ) k−1 k j6=k−1,k j k Q Remark: In the case when the polygon is Delzant, the integral inner normal vectors of two adjacent edges form a basis of Z2, so det(n n )2 = 1. k−1 k Proof. This result and its extension to higher dimension is due to Donaldson in [D2]. We perform the calculation globally in two dimensions to obtain the right constant; however, the main point is that the values at the vertices do not depend on the potential u. We have (nx)2 nxny f + i f + i i D2u = xx nxliny xy (nlyi)2 , (2.4) fxy +P ilii fyy + P lii ! so P P (nx)2(ny)2 nxnynxny detD2u = i j − i i j j l l l l i j i j i,j i,j X X f (ny)2 +f (nx)2 −2f nxny + xx i yy i xy i i +detD2f l i i X 1 = (nx)2(ny)2 −nxnynxny l l i j i i j j q k k "i6=j q6=i,j X(cid:0) (cid:1) Y Q + (f (ny)2 +f (nx)2 −2f nxny) l +detD2f l . xx i yy i xy i i j k # i j6=i k X Y Y The term in the brackets is the function 1/h. When evaluating h at the vertex v , both k l and l are zero, so only the terms from the first sum with i = k − 1, j = k, and k−1 k i = k, j = k −1 are nonzero, and therefore 1 = (nx )2(ny)2 −nx ny nxny + (nx)2(ny )2 −nxnynx ny l (v ) h(v ) k−1 k k−1 k−1 k k k k−1 k k k−1 k−1 q k k q6=k−1,k (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1) Y = nx ny −nxny 2 l (v ) k−1 k k k−1 q k q6=k−1,k (cid:0) (cid:1) Y =det(n n )2 l (v ). k−1 k q k q6=k−1,k Y (cid:3) Now we determine the restrictions on the Dirichlet boundary data. 6 DANIELRUBIN Lemma 2.2. Let u be a function which satisfies the Guillemin boundary condition (1.3) near the boundary of the polytope P. Set detD2u = 1/ϕ. Then Uniniϕ | = |n |2 (2.5) ni li=0 i where the limit is taken as x approaches any point on the edge away from the vertices. Proof. (ny)2 nxny f + i −f − i i nx Unknk = nxk nyk yy nlixny xy (nxl)i2 nky (cid:0) (cid:1) −fxy −P ilii fxx + P lii !(cid:18) k(cid:19) = (f (ny)2 +f (nx)2 −2f nxny) xx k yy k P xy k k P (ny)2 (nx)2 nxny + (nx)2 i +(ny)2 i −2nxny i i k l k l k k l i i i ! i i i X X X = (f (ny)2 +f (nx)2 −2f nxny) xx k yy k xy k k (ny)2 (nx)2 nxny + (nx)2 i +(ny)2 i −2nxny i i k l k l k k l i i i ! i6=k i6=k i6=k X X X since the terms with i = k cancel in the sum in parentheses. Also we have ϕ = h (n ·n ) l +D h l , (2.6) nk k j i nk j j i6=j j X Y Y so Unknkϕ | = h|n |2 l (f (ny)2 +f (nx)2 −2f nxny) nk li=0 k i xx k yy k xy k k " i6=k Y (ny)2 (nx)2 nxny + (nx)2 i +(ny)2 i −2nxny i i k l k l k k l i i i !# i6=k i6=k i6=k X X X = h|n |2h−1| k lk=0 = |n |2. k (cid:3) The boundary equation (2.8) was exploited in [LS] in connection with the variational approach to Abreu’s equation. In that context, this relation followed from the Euler- Lagrange equation satisfied by a minimizer, but in our context it follows directly from the Guillemin boundary conditions by computation as above. A very important consequence of the previous lemma is that, up to the values of the solutionu(x)atthevertices v ,··· ,v , theGuilleminboundaryconditions determine the 1 N boundary values of u. Indeed, in dimension two, the cofactor matrix entry Unn is equal to a constant multiple of the second tangential derivative along the edge. We may then THE MONGE-AMPE`RE EQUATION WITH GUILLEMIN BOUNDARY CONDITIONS 7 interpret this lemma as giving a second-order ODE on each edge for u. We parametrize the i-th edge, where {l (x) = 0}, by i x = v +tT . i i Lemma 2.3. Let u ∈ C2([0,L]) solve u = h(t) where h(t) is smooth and positive on tt t(L−t) [0,L]. Then u(t) = h(0)tlogt+h(L)(L−t)log(L−t)+v(t), (2.7) where v is smooth on [0,L]. The function u(t) is determined uniquely by its boundary values u(0) and u(L). Proof. By Taylor expanding h at 0, we see that h(0)tlogt accounts for the singularity there, and similarly at the other endpoint. What remains on the right-hand side is smooth, and can be integrated twice to obtain v. This proves the desired identity. The second statement is easy, since two solutions of this second order ODE must differ by an affine function of t. (cid:3) The following statement now follows readily from the previous two lemmas: Lemma 2.4. Let u ∈ C0(P) be a strictly convex function on P satisfying the equation (1.2) and the Guillemin boundary condition (1.3). Let α = u(v ) be the values of u at i i the vertices v , and define the function uˆ ∈ C0(∂P) as the unique solution on each edge i e of the equation i ∂2 uˆ = |n |2ϕ−1, uˆ(v ) = α , uˆ(v ) = α . (2.8) Ti i ni i i i+1 i+1 Then the function u is a solution of the Dirichlet problem, 1 detD2u = on P, u = uˆ. (2.9) ϕ(x) |∂P 3. Solution of Dirichlet Problem We turn now to the proof of Theorem 1.1, part (b). In view of Lemma 2.4, we shall define the desired solution u as the solution of the Dirichlet problem (2.9), where the Dirichlet data uˆ is specified by the values α and the function ϕ. i As a first step, we will first show the existence and uniqueness of generalized solutions to equations of this type, following closely the Perron’s method approach of Cheng-Yau. The only new difficulty is that our domain is a polygon, hence not strictly convex. This has consequences for the allowable boundary data and the regularity at the boundary. Recall the definition of an Alexandroff solution: Let u be a convex function on a domain Ω ∈ Rn. For each point x ∈ Ω, let B(x) = {p ,...,p } be the set of hyperplanes 1 n x = p x + b passing through (x,u(x)) and lying below the graph of u. To the n+1 i i function u we associate the measure µ(u), where µ(u)(E) = |B(E)|. Additionally we P define for ϕ ∈ C(Ω) the measure of u with weight ϕ to be µ (u,E) = ϕ(x)dµ(u,x) ϕ ZE 8 DANIELRUBIN for any Borel subset E of Ω. If µ (u) = µ where u is a convex function on Ω and µ is ϕ a Borel measure, then u is a generalized solution of detD2u = (1/ϕ)µ. In our equation, we take the Borel measure µ to be the ordinary Lebesgue measure. We make repeated use of the following two lemmas, which are now standard: Lemma 3.1. Let u be a sequence of convex functions defined on Ω which converges i uniformly on compact sets to a convex function u. Then µ(u ) converges to µ(u) weakly. i Lemma 3.2. Let u and u be two convex functions defined on a domain Ω with u = u 1 2 1 2 on ∂Ω and u ≥ u on Ω. Then µ(u ) ≥ µ(u ) 1 2 2 1 First we use a basic proposition taken directly from [CY], whose proof we include for convenience. Proposition 3.3. Let Ω be a polytope in Rn with vertices {v ,...,v }. Suppose ϕ ∈ C(Ω), 1 n ϕ ≥ 0, andfor allcompactsets K in Ω there is a constantc > 0 such thatinf ϕ(x) ≥ c. x∈K Let f be a function which is affine linear on ∂Ω, that is, f : ∂Ω → R such that f( λ v ) = λ a i i i i for any λ ≥ 0 and λ = 1, aX,...,a ∈ RX. Then for any Borel measure µ with i i 1 N compact support K contained in Ω and µ(Ω) < ∞, there exists a unique continuous convex function u on ΩP¯ such that µ (u) realizes µ and u = f on ∂Ω. ϕ Proof. First take µ to be a sum of point masses µ = m c δ (x), with c > 0. Let i=1 i xi i F denote the family of piecewise linear convex functions w with w = f on ∂Ω with P µ (w) ≤ µ (so the vertices of the polyhedron defined by the graph of w are a subset of G the {x }). F is non-empty since the convex hull of the data at the vertices is the graph i of a piecewise linear function equal to f on the boundary and with mass equal to 0. Set φ(w) = m w(x ). Then φ is bounded below in terms of infd(x ,∂Ω), infϕ(x ), i=1 i i i andµ(Ω)bytheAlexandroffmaximumprinciple. Inthetopologyofuniformconvergence, P F is compact, and φ is continuous, so φ achieves its minimum at some w¯ ∈ F. Then µ (w¯) = µ: If not, suppose the mass of µ (w¯) is strictly less than c at x . G ϕ 1 1 Then there exists ε > 0 such that the piecewise linear function wˆ obtained from w¯ by lowering its value at x by ε, that is, the function whose graph is the convex hull of 1 (x ,w¯(x )−ε),{(x ,w¯(x ))},{(v ,f(v ))}, also has mass less than µ. But φ(wˆ) < φ(w¯), 1 1 i i j j so we get a contradiction. Fora general Borel measure µ with compact support K, we let µ bea sequence of sums i of point masses converging weakly to µ, and u the sequence of piecewise linear functions i constructed as above with µ (u ) = µ . The functions u are uniformly bounded below ϕ i i i as before in terms of d(K,∂Ω), inf ϕ, µ(Ω), so they converge uniformly on compact K subsets to u with µ (u) = µ. Since also the u have bounded Lipschitz norm in terms ϕ i of the boundary data and d(K,∂Ω), inf ϕ, µ(Ω), u has bounded Lipschitz norm and K u = f on ∂Ω. (cid:3) Now we want to solve with more general boundary data. Since we remain in the setting of polygons, which are not strictly convex, we must insist that the boundary data is convex on each face. THE MONGE-AMPE`RE EQUATION WITH GUILLEMIN BOUNDARY CONDITIONS 9 Proposition 3.4. Let Ω be a polygon in R2. Let f : ∂Ω → R be convex on each edge and continuous, and ϕ, µ as in Proposition 3.3. Then there is a unique continuous convex function u on Ω¯ such that µ (u) realizes µ and u = f on ∂Ω. ϕ Proof. We approximate the solution of this problem with the solutions of Proposition 3.3 by taking a sequence of sets of vertices where A = {v ,...,v }, the set of vertices 1 1 N of Ω, and each set A contains the midpoint of any vertices from the previous set A n n−1 lying on the same edge of Ω. The same proof shows that for each set of vertices A , the n Dirichlet problem can be solved in Ω for a continuous convex function u with boundary n data equal to f(x ) at each point x of A and linear on the edges in between, since we i i n can still form the non-empty family of piecewise linear convex functions in Ω matching the boundary data with mass less than a sum of point masses. Theu areuniformlyboundedbelowanddecreasing, andthusconverge toacontinuous n solution u with u = f on ∂Ω. Note that the convergence is not necessarily Lipschitz in (cid:3) the corners since the boundary data need not be Lipschitz there. Now we must allow the measure µ to have support up to the boundary. Theorem 3.5. Let P = ∩{l > 0} be a polygon in R2. Let f : ∂P → R be continuous i and convex, with second tangential derivatives f < C/d, where d is the distance to tt the nearest vertex. Let ϕ be a smooth function such that there exist positive constants a,A where a l ≤ ϕ ≤ A l , and µ a finite Borel measure. Then there exists a i i unique continuous convex function u on P such that µ (u) realizes µ. Moreover, for all ϕ Q Q 0 < α < 2/(N +1), u ∈ Cα(P), with norm bounded in terms of C, µ(P), a, and A. Proof. Let {h } be an increasing sequence of cutoff functions with compact support in i P such that 0 ≤ h ≤ 1 and h is identically equal to 1 on any compact subset of P for i i i sufficiently large. From the previous proposition, we have a sequence of functions u i such that u = f on ∂P and µ (u ) realizes h µ. By the lemma, the u are a decreasing i ϕ i i i sequence of functions. To establish the existence of a limit with the stated boundary regularity, we must find a lower barrier. This step is more difficult for a polygon than in the case of a uniformly convex domain because of the lack of a C2 convex defining function. α Lemma 3.6. For 0 < α < 2/(N + 1), the function φ(x) = l (x) is strictly 1≤i≤N i concave in P. (cid:0)Q (cid:1) ˜ Assuming the lemma, we let f be any smooth extension of the boundary values to the ˜ interior bounded by the values of f and with Hessian bounded by f and let v(x) = f + tt A(−φ(x)). Then v = f on ∂P with detD2v ∼ l2α−2 near l = 0 and detD2v ∼ (l l )2α−2 i i i j near the corner l = l = 0, so v ≤ u for each i. Therefore u converges uniformly on i j i i compact subsets to a continuous convex function u on P such that u = f on ∂P and µ (u) realizes µ, and u ∈ Cα(P¯). (cid:3) ϕ Proof of Lemma 3.6. Notethatforasinglecorner, onecaneasilyseebydirectcalculation of the Hessian that the function ((y + λx)y)α is concave for 0 < α ≤ 1/2 and strictly concave for 0 < α < 1/2 in the region {y + λx > 0} ∩{y > 0}. For the barrier in the 10 DANIELRUBIN whole polygon, we show that the function φ(x) is strictly concave on any line segment contained in P. When restricted to a line parametrized by t, we have α φ(x) = (a +b t) , i i ! 1≤i≤N Y and therefore 2 N N 2 b b i i φ = αφ α − tt  a +b t a +b t  i=1 i i ! i=1 (cid:18) i i (cid:19) X X  N N  2 b b b i j i = αφ α − a +b ta +b t a +b t i,j=1(cid:18) i i j j (cid:19)! i=1 (cid:18) i i (cid:19) ! X X N 2 2 N 2 α b b b i j i ≤ αφ + − 2 a +b t a +b t a +b t i,j=1 (cid:18) i i (cid:19) (cid:18) j j (cid:19) ! i=1 (cid:18) i i (cid:19) ! X X N 2 (N +1)α b i = αφ −1 , 2 a +b t (cid:18) (cid:19) i=1 (cid:18) i i (cid:19) X which is negative if 0 < α < 2 . N+1 (cid:3) Hence for every choice of values at the vertices {α }, there exists a unique continuous k convex solution u to the Dirichlet problem (2.9), which is Hölder continuous of exponent α for any α < 2/(N + 1) at the boundary. Restricting this solution to any uniformly convex subdomain that does not touch the boundary, we have a solution of a Monge- Ampère equation with uniformly bounded right-hand side, so by the results of Cafarelli and Gutiérrez [CG], the solution is in fact smooth in the interior. 4. Behavior Near the Interior of an Edge Now we investigate the behavior of the solution u at the boundary near an edge and away from the vertices. We take our edge to be a segment of {y = 0} containing an interval around (0,0). Our goal is to show that in a small half-disc B+(0), u = ylogy+f, r where f ∈ C∞(B+(0)). The main technique is the partial Legendre transform as in [DS], r which is useful in dimension two, where the transformed function satisfies a quasilinear equation. (Another way to understand why the dimension two case is simpler, without reference to the partial Legendre transform, is as follows: In dimension two, the second tangential derivative u a solution to linear equation with 0 right-hand side, so it is xx possible to obtain a positive lower bound for u .) xx As a model, consider the degenerate Monge-Ampère equation 1 1 detD2u = in R2 , u| = x2. (4.1) y y>0 y=0 2